 In this lecture, we will further discuss about the meaning of negative temperature and how negative temperature can affect the various quantities that we have discussed so far that is partition function, internal energy and entropy. We have already discussed that under what conditions this temperature can be negative and if you recall the previous lecture, we consider two level system and in this lecture also our discussion is also going to be on two level system. In two level system, we discussed that if the population of upper level exceeds that of the lower level then the temperature will be negative. Now, let us see how it affects the partition function. Partition function q will be equal to g 0 plus g 1 exponential minus e upon k t for a two level system right. If there is a degeneracy or if the system is non-degenerate let us say if I for non-degenerate systems non-degenerate non-degenerate energy levels then q can be written as 1 plus exponential minus e upon k t. Our now goal is to discuss t varying from plus 0 to infinity and t varying from minus 0 to infinity. Remember that the formula does not restrict temperature less than 0. First let us talk about t greater than 0. We talk in general about extremes but the variation over entire temperature range is shown in this figure. Let us first talk about t approaching a value of plus 0. If t is plus 0 then q will be equal to 1 plus exponential minus infinity that means q will approach a value of 1 and that is what you see over here. Now let us take the other extreme when t approaches plus infinity then q will be equal to 1 plus exponential minus e over k into infinity is infinity this becomes that means q approaches a value of 2. Either it will approach a value of 2 or it will be approaching a value of g 0 plus g 1. So, that is why here the variation is not ended at 2 but it can depend upon degeneracies. If both are need non-degenerate then it will be approaching a value of 2. So, we have explained this variation and this is t is equal to 0 and positive onwards. This discussion we have had in other numerical problems earlier. Now let us talk about t approaching a value of minus 0 from negative side then q is equal to 1 plus exponential minus 0 minus minus becomes positive I am talking about here substituting here minus 0 minus minus becomes positive and what I have is minus e upon k t is 0. This is exponential since minus minus becomes positive this will be. So, that means q becomes very high in fact moving towards infinity and that is what you see over here. It is rising very sharply when temperature is approaching minus 0 then q is increasing very sharply. Let us now discuss the next one when t approaches minus infinity then q will be approaching what 1 plus exponential plus e upon infinity which is 1 plus exponential 0 that means q will approach a value of 2 and that is what you see over here. So, for a two level system see how the partition functions varies with temperature when you consider positive side of the temperature or negative side of the temperature, but the point that is to be noted over here that this is there is a discontinuity it is not continuous at t is equal to 0 there is a discontinuity. Let us talk about the internal energy also for a two level system the partition function is like this and internal energy is also showing a discontinuity. The way we have shown it for partition function same way we can show for the internal energy let us do that since we are talking about two level system and let me take the simplest 1 0 and e. So, what I have is q is equal to 1 plus exponential minus beta e this is q for this system u minus u 0 will be equal to minus n by q del q by del beta at constant volume that we know. So, what I have now is u minus u 0 is equal to minus n divided by 1 plus exponential minus beta e into derivative of this exponential minus beta e into minus e. So, I have u minus u 0 is equal to n times e into exponential minus beta e negative negatives are consumed to become positive sign 1 plus exponential minus beta e. I can rewrite this u minus u 0 if I multiply and divide by exponential beta e then I get this over exponential beta e plus 1. This is how the internal energy will vary with the temperature. We will discuss it little bit more. So, we have derived now this expression that internal energy is given by n e divided by exponential beta e plus 1. Let us discuss the conditions when beta approaches a value of infinity when beta approaches infinity beta is equal to 1 over k t remember beta is 1 over k t when beta approaches infinity means temperature approach is 0 temperature approach is 0. So, then you will approach a value of 0 because when beta approaches infinity temperature approach is 0 u approaches 0. And when do we have the total energy equal to 0 when all the molecules are in the ground state that means this situation u approaches 0 as beta approaches infinity that is as t approaches 0 when only the lower state is occupied. And you can notice over here also when t approaches 0 then internal energy approaches a value of 0. Now, let us consider the next extreme when beta approaches a value of 0. Let us consider the next extreme when beta approaches a value of 0 that means t is approaching a value of infinity in that case what will be u minus u 0 u minus u 0 will be n e divided by exponential beta is 0 tending to 0 0 plus 1 which is 0.5 times n e. And that is what you are seeing over here when temperature becomes very very high the value of u by n e is approaching 0.5. Let us now consider the other case when beta approaches minus infinity that means t will approach a value of minus 0 then what happens then u will be what u minus u 0 will be equal to t is minus 0 that means you have n e divided by when beta is minus infinity that means 1 over infinity which becomes 0 right. I can write here beta is 0 that means t is minus exponential minus infinity plus 1 this is equal to n times e that means u divided by n e will approach a value of 1 that is what you see here. The next one when beta approaches minus 0 that means t approaches minus infinity then what we have then u minus u 0 is equal to n e over exponential 0 in fact minus 0 plus 1 exponential 0 or exponential minus 0 is 1 which is 0.5 n e that is what you see over here. So, the variation is from 1 to 0.5 and here is from 0 to 0.5 it is another way of looking at that when the temperature becomes very high then both the states are equally likely to be populated. So, therefore the value will start from 0 to half u by n e will become half. Let us read the comments note that u approaches 0 and beta approaches infinity that is as t approaches 0 when only the lower state is occupied and u approaches n e as beta approaches minus infinity that is t approaches minus 0 we see that a state with t is equal to minus 0 here you see is hotter than the 1 t is equal to plus 0. So, you see the meaning of t negative minus 0 is hotter than the 1 with t plus 0 and also note down that in this figure as you see what we see is that q and u show sharp discontinuities we see sharp discontinuities on passing through 0 and t is equal to plus 0 corresponding to all populations in the lower state is distinct from t is equal to minus 0 where all the population is in the upper state. This is consistent with when we introduce that when the temperature can be negative for a two level system this is how the internal energy will change with temperature on both sides of t equal to 0. Let us further discuss now what does it mean when t is equal to 0 or t greater than 0 or t greater than 0 or t less than 0 we notice there is a sharp discontinuity we also notice that when t is positive that means we are talking about when t when you are approaching t is equal to plus 0 you are talking about cooling cold, but when we are talking about t negative and when t is approaching minus 0 you can see here the system here is much hotter than the 1 which is equal to t plus 0 are these discontinuities carried to when we talk about entropy we will see about entropy here and we can use the same expression that we derived for u minus u 0 what we have s is equal to u minus u 0 we had n e over I will retain t here and you have exponential beta e plus 1 this is u minus u 0 by t plus n k log q is 1 plus exponential minus beta e this is for a two level system we are talking about two level system here we see some continuity let us say when I talk about t approaching 0 then beta is approaching infinity beta is 1 over k t and then what is the value of s approaching if beta is infinity that then you know since infinity is appearing in denominator this term first term this disappears and if I put beta as infinity exponential minus infinity is 0 and n k log 1 in that case is 0 I repeat when beta is 0 then what is the value of beta approaching infinity there is infinity in the denominator so this disappears and beta is infinity exponential minus infinity is 0 so I have n k log 1 log 1 is 0 that means in this case s approaches 0 and you can notice over here and when t approaches infinity then beta approaches 0 then what do we have when beta approaches 0 t approaches infinity then we have t here in the denominator so this term disappears and beta approaches 0 then this term becomes 1 exponential minus beta e becomes 1 that means in that case your s will approach a value of n k log 2 and log 2 is 0.693 and that is what you observe the value over here it approaches a value of 0.693 similarly you can talk about when t approaches infinity approaches minus infinity minus infinity anyway when infinity comes in the denominator the first term will become 0 and t is minus infinity that means beta is approaching minus 0 so therefore when you put beta is minus 0 exponential 0 and here what you have s will approach n k log 2 and that is what you observe over here. Therefore the entropy of the system is 0 on either side of t equal to 0 and rises to n k log 2 as t approaches plus minus infinity at t plus 0 only one state is accessible and that is the lower state. Now note down that the functions that that we just plotted for example if I start from partition function the partition function here was plotted against k t by epsilon and in the next one internal energy was plotted against k t by epsilon and even entropy when we talked about was plotted against k t by epsilon and if instead of k t by epsilon I choose to plot against epsilon upon k t then the variation is going to be as shown in these figures ok. So the behavior of the plot obviously will depend upon how we represent it. Now we can get a little bit more insights into the dependence of thermodynamic properties on temperature that can be obtained by noting the thermodynamic result. The thermodynamic result that I am talking about here is t is equal to del u by del s at constant volume from where this comes first law of thermodynamics du is equal to dq plus dw and you remember that the fundamental equation what was that dq is equal to tds that means I will write du is equal to tds minus pdv dw is minus pdv. So therefore I can connect del u del s with temperature I can have a pure thermodynamic definition of temperature that means the pure thermodynamic definition of temperature thus becomes t is equal to del u by del s at constant volume it comes from this you fix the volume constant and t becomes del u by del s at constant volume. If you plot entropy in the units of s by n k against internal energy it will show this kind of behavior and that you can obtain by using the appropriate expressions. But what is more important here is to recognize that when s is plotted against u for a two level system we see that entropy rises as the energy is supplied to the system provided t is greater than 0. However the entropy decreases as the energy is supplied at t less than 0. So what does it mean that if you supply energy del u is positive and the entropy decreases del s is negative that means we are talking about negative temperature. Physically the increase in entropy for t greater than 0 corresponds to increasing accessibility of the upper state we have discussed it earlier and the decrease for t less than 0 corresponds to shift towards population of the upper state alone as more energies packed into the system. Same thing we have discussed in the earlier slides recall the second law of thermodynamics the efficiency of heat engines which of course is a direct consequence of second law is given by 1 minus t cold by t hot fine that means if the temperature of the cold reservoir is negative if t is negative then this negative negative becomes plus then the efficiency of engine may be greater than 1 if we talk about the negative temperature. This condition corresponds to amplification of signals achieved in lasers as I said that this advanced level you will study later on. Alternatively what is the meaning of efficiency greater than 1 it means that heat can be converted completely into work provided the heat is withdrawn from a reservoir which is at a negative temperature. Now if both the reservoirs are at negative temperatures then then efficiency is less than 1 as in the thermal equilibrium because if both are negative negative negative become positive and overall negative sign remains over there. Now this all discussion may lead us to conclusion which is accepted and as mentioned in several text books that the third law may require a slight amendment on account of discontinuity of populations across t equal to 0 based upon our discussion that how these various thermodynamic quantities vary when t is on the positive side of 0 or on the negative side of 0 and what the amendment is suggested is that it is impossible in a finite number of steps to cool any system down to plus 0 or to heat any system above minus 0. So, what we have discussed in this lecture is that when negative temperature is possible what is the meaning of negative temperature and how the negative temperature can affect the variation of thermodynamic properties as a function of temperature. How to achieve this negative temperature or under what conditions what techniques can be used to describe this negative temperature are beyond the scope of current discussion you will study at a later level, but I am sure that the meaning of negative temperature is clear. Thank you very much.